In some cases, recognizing some common patterns in the quadratic equation will help you
to factorize the quadratic. For example, the quadratic equation could be a Perfect Square Trinomial (Square of a Sum or Square of a Difference) or Difference of Two Squares.

Perfect Square Trinomial (Square Of a Sum)

A square of a sum or perfect square trinomial is a type of quadratic equations of the form:

x2 + 2bx + b2 = (x + b)2

Example 1:

x2 + 2x + 1 = 0

(x + 1)2 = 0

Example 2:

x2 + 6x + 9 = 0

x2 + 2(3)x + 32 = 0

(x + 3)2 = 0

Perfect Square Trinomial (Square of a Difference)

A square of difference is a type of quadratic equations of the form:

x2 – 2bx + b2 = (x – b)2

Example 1:

x2 – 2x + 1 = 0

(x – 1)2 = 0

Example 2:

x2 – 6x + 9 = 0

x2 – 2(3)x + 32 = 0

(x – 3)2 = 0

x – 3 = 0 ⇒ x = 3

This video provides examples of how to factor and solve quadratic equations when the equations are perfect square trinomials.

Perfect Square Trinomials

One special case when trying to factor polynomials is a perfect square trinomial. Unlike a difference of perfect squares, perfect square trinomials are the result of squaring a binomial. It's important to recognize the form of perfect square trinomials so that we can easily factor them without going through the steps of factoring trinomials, which can be very time consuming.

Factor perfect square trinomial

Difference of Two Squares

A difference of two squares is a type of quadratic equations of the form:

(a + b)(a – b) = a2 – b2

Example:

x2 – 25 = 0

x2 – 52 = 0

(x + 5)(x – 5) = 0

We get two values for x:

Be careful! This method only works for difference of two squares and not for the sum
of two squares: a2 + b2 ≠
(a + b)(a – b)

The following videos explain how to factor a difference of squares.

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